You can entangle as many particles as you want -- however, there is a property of quantum entanglement called the monogamy of entanglement. This could be interpreted as a limit on the extent to which systems of many particles can be inter-entangled.

Generally, energy within a system will tend to distribute until thermodynamic equilibrium is reached. But for a lot of systems that we study, it's a fair assumption that it's coupled to an environment that acts as a large, empty energy sink. So that sink will tend to take all the energy until the system we're interested in ends up in its lowest energy configuration.

For example, an electron orbiting an atom is coupled with the electromagnetic field, which is pretty empty for most situations. So if it's in an excited energy level, it will tend to dump that energy out as a photon until it reaches the ground state. But if the electromagnetic field is locally very active, with photons whizzing around everywhere, this approximation fails and you have to treat the electron's energy level statistically (like in a laser).

Another factor is friction, which in very abstract terms could be defined as the tendency for energy to fall out of macroscopic degrees of freedom towards microscopic. That's what ultimately makes a stirred fluid stop sloshing around, with the energy being dissipated into smaller and smaller vortices until it simply becomes heat.

> Is there a chance that the r in the equation in this case would actually represent the distance between the surface of the sphere and the point, rather than the center?

The charges for a conducting sphere distribute themselves uniformly on the surface of the sphere. Each little element of charge on the surface contributes an infinitesimal amount of the final electric field. To calculate the final field, you need to (vector) integrate the contributions from all these elements. So you can't just use kq/r^2 but with r the "altitude" of the test charge.

If you do this calculation, you'll find that it actually can apply E = kq/r^(2) with r the distance to the centre of the sphere -- the uneven contributions cancel out. From the outside, a spherical shell of uniform charge looks exactly like a point charge at its centre.

This is actually true for any company spherical charge distribution, and you can prove it very elegantly using Gauss's law.

It's quite complicated, but I'll try to give a brief answer --

Maxwellian electrodynamics is a classical field theory. When you canonically quantise such a theory, you find that a (sort of*) conserved, discrete quantity pops out, which can be interpreted as "particle number".

This is in line with observations from a century back around black-body radiation that appeared to show quantisation of energy levels.

It's also a satisfying interpretation because certain calculations in QFT have combinatorial properties which allow them to be represented using Feynman diagrams (which you've probably heard of). Together with the path integral formulation, you get a really useful picture of the physics of scattering. But this is mired in complications (renormalisation, confinement (not a problem for photons but it is for quarks), divergences around every corner, ...).

As a side note, instead of starting from the field theory, you can build up a quantum field theory by starting from a particle-based theory; a common approach for effective theories in condensed matter theory, because it's much more tractable mathematically. The fact that QFT unifies classical many-particle and field theories is an advanced form of the "wave-particle duality" you might have heard of.

*: conserved in the free theory

I'm glad you liked my comment! I do have a blog, which I'll link here (assuming there aren't any rules against it -- I can take down my comment otherwise). It's pretty empty at the moment though. I've meant to put down all my thoughts about physics and intuition in one place at some point, but I just haven't gotten around to it. If you liked it, maybe there is an audience for it!

I'm very glad you found my reply useful!

If you're set on continuing to teach yourself physics (which I think is a very good, though time-consuming idea), I'd start by making sure you're on top of your high school/A-Level maths and physics (KhanAcademy is a great place for this), and then move onto some first-year university introductory textbooks. You don't have to read them back to front -- start with the first chapter, take your time, do the exercises, and when you get bored switch to a different book. (I really like Griffiths' textbooks, but YMMV.)

A good search term is "introduction to [topic]" or "introductory [topic] textbooks". Good topics to start with would be classical mechanics, electrodynamics, and quantum mechanics. You could then move onto special relativity and statistical physics (my favourite!).

Before I try to answer your question, I think you're suffering from a (very common!) issue with your approach to physics. I've written a lot about this topic on Reddit before, so I'll link some comments: see here, here, here, and here. Though it's not directly connected to your question, I hope that these can be helpful!

Now to get to your question. I'd start by not trying to understand photons right off the bat. It'll be much easier to first try to understand the electromagnetic field classically, and then try to integrate quantum effects into your understanding. The relationship between photons and the classical field is very tricky, and I find it's the cause of much confusion for physics enthusiasts and students.

In classical/Maxwellian electrodynamics, the electric and magnetic fields are vector fields. That is to say, they are mathematical functions which assign a three-dimensional vector to each point in space. These must obey Maxwell's equations (that's part of the theory).

Wave solutions to Maxwell's equations look something like this. (Shockingly, I couldn't find a clean diagram like this on the Internet, so I had to make one in Paint.) Of course, I can't show you every single vector attached to every single point in space on my diagram, because there's an infinite amount of them, so I've just drawn them for a select grid of points. I'm using the notation where vectors going into the plane are drawn with a cross, and those sticking out of the plane with a dot.

The size of the arrows is meant to represent the relative magnitudes of the fields as they evolve through space, but the actual lengths on the diagram don't matter -- the electric and magnetic fields don't have units of distance.

Hopefully this helps you visualise the idea of wavelength -- let me know if you have follow-up questions!

> How the temperature relates to other state variables is different (and difficult!) in liquids but that doesn't apply to kinetic energy

Is there a simple explanation for why this is the case? Given the presence of intermolecular potentials (which are not quadratic terms), I wouldn't expect equipartition to hold. Is the argument that this effect is negligible, and if so, how does one argue that it is?

Furthermore, does your calculation account for vibrational and rotational modes?

If you could point me to sources that cover these questions, I'd be very grateful. Thanks.

In what context are you seeing the term "quantised vector space"?

In any case, you may find it interesting to read about vector spaces over finite fields.